nLab
symplectomorphism group

Idea

Traditional

For $(X,\omega )$ a symplectic manifold, the symplectomorphism group

$$\mathrm{Diff}(X,\omega )\hookrightarrow \mathrm{Diff}(X)$$Diff(X,\omega) \hookrightarrow Diff(X)

is the subgroup of the diffeomorphism group of $X$ on the diffeomorphisms.

In higher symplectic geometry

Analogous constructions apply when symplectic manifolds are generalized to n-plectic infinity-groupoids: for $(X,\omega )$ an n-plectic manifold, and $n$-plectomorphism is a diffeomorphism $\varphi :X\to X$ that preserves the $n$-plectic form ${\varphi }^{*}X\simeq X$.

Examples

• The linear part of the 2-plectomorphism group/3-plectomorphism group of the Cartesian space ${ℝ}^7$ equipped with its associative 3-form $\omega =lanlge(-),(-)\times (-)⟩$ is the exceptional Lie group G2. See there for more details.

Related concepts

A further subgroup is that of Hamiltonian symplectomorphisms. The group extension of that whose elements are pairs consisting of a Hamiltonian diffeomorphism and a choice of Hamiltonian for this is the quantomorphism group.

The Lie algebra of the symplectomorphism group is that of symplectic vector fields.

higher and integrated Kostant-Souriau extensions

(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for $𝔾$-principal ∞-connection)

$$(\Omega 𝔾)\mathrm{FlatConn}(X)\to \mathrm{QuantMorph}(X,\nabla )\to \mathrm{HamSympl}(X,\nabla )$$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)

$n$	geometry	structure	unextended structure	extension by	quantum extension
$\mathrm{\infty }$	higher prequantum geometry	cohesive ∞-group	Hamiltonian symplectomorphism ∞-group	moduli ∞-stack of $(\Omega 𝔾)$-flat ∞-connections on $X$	quantomorphism ∞-group
1	symplectic geometry	Lie algebra	Hamiltonian vector fields	real numbers	Hamiltonians under Poisson bracket
1		Lie group	Hamiltonian symplectomorphism group	circle group	quantomorphism group
2	2-plectic geometry	Lie 2-algebra	Hamiltonian vector fields	line Lie 2-algebra	Poisson Lie 2-algebra
2		Lie 2-group	Hamiltonian 2-plectomorphisms	circle 2-group	quantomorphism 2-group
$n$	n-plectic geometry	Lie n-algebra	Hamiltonian vector fields	line Lie n-algebra	Poisson Lie n-algebra
$n$		smooth n-group	Hamiltonian n-plectomorphisms	circle n-group	quantomorphism n-group

(extension are listed for sufficiently connected $X$)